Flux in tilted potential systems: negative resistance and persistence

TL;DR

This paper develops a graph-theoretic and Morse-theoretic framework to analyze steady-state fluxes in Brownian systems on manifolds, explaining phenomena like negative resistance through rigorous mathematical methods.

Contribution

It introduces a novel graph-theoretic formula for small-noise flux asymptotics and connects flux to entropy production and persistent homology in gradient-like systems.

Findings

Derived a graph-theoretic formula for flux asymptotics

Connected flux to entropy production rate in gradient systems

Provided a rigorous explanation for negative resistance phenomenon

Abstract

Many real-world systems are well-modeled by Brownian particles subject to gradient dynamics plus noise arising, e.g., from the thermal fluctuations of a heat bath. Of central importance to many applications in physics and biology (e.g., molecular motors) is the net steady-state particle current or "flux" enabled by the noise and an additional driving force. However, this flux cannot usually be calculated analytically. Motivated by this, we investigate the steady-state flux generated by a nondegenerate diffusion process on a general compact manifold; such fluxes are essentially equivalent to the stochastic intersection numbers of Manabe (1982). In the case that noise is small and the drift is "gradient-like" in an appropriate sense, we derive a graph-theoretic formula for the small-noise asymptotics of the flux using Freidlin-Wentzell theory. When additionally the drift is a local gradient sufficiently close to a generic global gradient, there is a natural flux equivalent to the entropy production rate -- in this case our graph-theoretic formula becomes Morse-theoretic, and the result admits a description in terms of persistent homology. As an application, we provide a mathematically rigorous explanation of the paradoxical "negative resistance" phenomenon in Brownian transport discovered by Cecchi and Magnasco (1996).